Problem: Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $p \neq 0$. $k = \dfrac{5p^2 + 25p}{2p^3 - 8p^2 - 90p} \times \dfrac{p^2 - 19p + 90}{5p + 45} $
Answer: First factor out any common factors. $k = \dfrac{5p(p + 5)}{2p(p^2 - 4p - 45)} \times \dfrac{p^2 - 19p + 90}{5(p + 9)} $ Then factor the quadratic expressions. $k = \dfrac {5p(p + 5)} {2p(p - 9)(p + 5)} \times \dfrac {(p - 9)(p - 10)} {5(p + 9)} $ Then multiply the two numerators and multiply the two denominators. $k = \dfrac {5p(p + 5) \times (p - 9)(p - 10) } { 2p(p - 9)(p + 5) \times 5(p + 9)} $ $k = \dfrac {5p(p - 9)(p - 10)(p + 5)} {10p(p - 9)(p + 5)(p + 9)} $ Notice that $(p - 9)$ and $(p + 5)$ appear in both the numerator and denominator so we can cancel them. $k = \dfrac {5p\cancel{(p - 9)}(p - 10)(p + 5)} {10p\cancel{(p - 9)}(p + 5)(p + 9)} $ We are dividing by $p - 9$ , so $p - 9 \neq 0$ Therefore, $p \neq 9$ $k = \dfrac {5p\cancel{(p - 9)}(p - 10)\cancel{(p + 5)}} {10p\cancel{(p - 9)}\cancel{(p + 5)}(p + 9)} $ We are dividing by $p + 5$ , so $p + 5 \neq 0$ Therefore, $p \neq -5$ $k = \dfrac {5p(p - 10)} {10p(p + 9)} $ $ k = \dfrac{p - 10}{2(p + 9)}; p \neq 9; p \neq -5 $